On the rigidity side,
the magnetic helicity conservation for the 3D ideal MHD was proved
in [1]}, [2]}, [3]} and [4]},
respectively,
for the critical spaces \(L^{3}_{t,x}\)  and \(L^3_tB^{\alpha }_{3,{\infty },x}\)  with \({\alpha }>0\) .
The corresponding rigidity results for the total energy conservation
were proved in [4]}.
On the flexible side,
weak solutions with non-trivial energy
and vanishing magnetic helicity were first given in [6]},
by embedding the ideal MHD into the \(2\frac{1}{2}\) D Euler flow via a symmetry assumption.
Wild solutions with compact support in space-time were first constructed by
Faraco-Lindberg-Székelyhidi [7]},
via the \(L^{\infty }_{t,x}\)  convex integration scheme.
The constructed solutions in [7]}
violate the energy conservation, while the magnetic helicity is still conserved
and thus vanishes due to the compact support in time of solutions.
